While the other steps of alg:ft are relatively straightforward, we explain the success-check step of alg:ft.
After applying the recovery operation, all the cluster state stabilizers are guaranteed to be satisfied.
Therefore, error correction is successful if the product of the qubit error and the recovery operator is a stabilizer (logical identity operator) and error correction fails if the product of the qubit error and the recovery operator is a non-trivial logical operator.
Such operators anti-commute with at least one of the correlation surfaces of the cluster state [1]}.
fig:corr-surf shows the \(X\)  correlation surface (the \(Z\)  correlation surface is analogous).
To summarize, if the product of the qubit error and the recovery operator anti-commutes with either correlation surface, then error correction has failed.
